Build a charcoal fire or preheat a gas grill. In a deep 10-inch cast-iron skillet, melt the butter over indirect medium-low heat. Stir in 1/4 cup sugar and cook until browned, about 5 minutes. Add a pinch salt, 1/4 teaspoon cinnamon and the sliced peaches. Cook until caramelized, about 5 minutes. Arrange the fruit in an even layer in the pan and let cool.
In a medium bowl, toss the chopped peaches with the remaining 2 cups sugar and 1/4 teaspoon cinnamon. Sift together the flour, baking soda and remaining 1/2 teaspoon salt. In a large bowl, whisk together the eggs and oil, then add the vanilla. Gradually stir the flour mixture into the egg mixture until well mixed. Stir in the chopped peach-sugar mixture. Fold in the hazelnuts. Let the batter stand for 15 minutes. Stir the batter again (it will be very thick) and spread over the cooled fruit in the skillet.
Place the skillet in the center of the grill and bake over indirect medium-low heat, with the lid down, until a toothpick inserted in the center comes out clean, 1 1/4 to 1 1/2 hours. Let cool for 15 minutes. Invert onto a platter and serve warm.